144 research outputs found
Cone Conditions and Covering Relations for Topologically Normally Hyperbolic Invariant Manifolds
We present a topological proof of the existence of invariant manifolds for
maps with normally hyperbolic-like properties. The proof is conducted in the
phase space of the system. In our approach we do not require that the map is a
perturbation of some other map for which we already have an invariant manifold.
We provide conditions which imply the existence of the manifold within an
investigated region of the phase space. The required assumptions are formulated
in a way which allows for rigorous computer assisted verification. We apply our
method to obtain an invariant manifold within an explicit range of parameters
for the rotating H\'enon map
Transition Tori in the Planar Restricted Elliptic Three Body Problem
We consider the elliptic three body problem as a perturbation of the circular
problem. We show that for sufficiently small eccentricities of the elliptic
problem, and for energies sufficiently close to the energy of the libration
point L2, a Cantor set of Lyapounov orbits survives the perturbation. The
orbits are perturbed to quasi-periodic invariant tori. We show that for a
certain family of masses of the primaries, for such tori we have transversal
intersections of stable and unstable manifolds, which lead to chaotic dynamics
involving diffusion over a short range of energy levels. Some parts of our
argument are nonrigorous, but are strongly backed by numerical computations
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